1. Lecture 21: B Spline Curve
CS552: Computer Graphics
Lecture 21: B Spline Curve

2. Recap Bezier curve Properties Rendering De Casteljau's Algorithm
Subdividing Bezier Curve
Continuity of curve
𝐶 0
𝐶 1
𝐶 2

3. Objective After completing this lecture, students will be able to
Explain the issues with Bezier curve representation
Explain the advantage of B spline curve
Calculate the B-spline basis of different degrees and knot intervals

4. Bezier Curves: Issues No local control
Degree of curve is fixed by the number of control points

5. B Spline Each control point has a unique basis function
Local control is facilitated

6. B spline Curves
The user supplies: the degree p, n+1 control points, and m+1 knot vectors
Write the curve as:
The functions Nip are the B-Spline basis functions
B-Spline Animation

7. B Spline Basis The domain is subdivided by knots, and
Basis functions are not non-zero on the entire interval.
Some knot spans may not exist (Repeat)
Simple / Multiple Knots
Uniform/ Non-Uniform Knots
The i-th B-spline basis function of degree p
𝑁 𝑖 0 𝑡 = 1, 𝑡 𝑖 ≤𝑡≤ 𝑡 𝑖+1 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
B-Spline Basis Plots
𝑁 𝑖 𝑝 𝑡 = 𝑡− 𝑡 𝑖 𝑡 𝑖+𝑝 − 𝑡 𝑖 𝑁 𝑖 𝑝−1 𝑡 + 𝑡 𝑖+𝑝+1 −𝑡 𝑡 𝑖+𝑝+1 − 𝑡 𝑖 𝑁 𝑖+1 𝑝−1 𝑡
Cox-de Boor recursion formula

8. B Spline Basis: Observations 1
Non-zero domain of a basis function
Basis function 𝑵𝒊,𝒑(𝒖) is non-zero on [𝒕𝒊, 𝒕 𝒊+𝒑+𝟏 )

9. B Spline Basis: Observations 2
Influence of the basis function coefficients 𝑡
𝑡 𝑖
𝑡 𝑖+1
𝑡 𝑖+𝑝
𝑡 𝑖+𝑝+1
𝑡 𝑖+𝑝 − 𝑡 𝑖
𝑡− 𝑡 𝑖
𝑡 𝑖+𝑝+1 −𝑡
𝑡 𝑖+𝑝+1 − 𝑡 𝑖+1
Linear combination of two intervals, where both are linear in 𝑢

10. Example Suppose the knot vector is U = { 0, 0.25, 0.5, 0.75, 1 }.
Hence, m = 4 and u0 = 0, u1 = 0.25, u2 = 0.5, u3 = 0.75 and u4 = 1. 
Degree
Basis Function
Range
Equation 1
𝑁 0 0
𝑁 1 0
𝑁 2 0
𝑁 3 0
𝑁 0 1
𝑁 1 1
𝑁 2 1

11. Thank you
Next Lecture: B-Spline Curve